The class of DCT-graphs is a common generalization of the classes of almost claw-free and quasi claw-free graphs. We prove that every even (2p + 1)-connected DCT-graph G is p-extendable, i.e. every set of p independent edges of G is contained in a perfect matching of G. This result is obtained as a corollary of a stronger result concerning factor-criticality of DCT-graphs.